prv_lec3 - Probability Random Variables Chapter 3 Hyung Il...

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Probability & Random Variables Chapter 3 Hyung Il Koo [email protected] Department of Electrical Engineering Ajou University, Korea Spring 2016
3.1 Expectation Expectation the name given to the process of averaging when a random variable is involved E [ X ] or ¯ X for a random variable X the mathematical expectation of X the expected value of X the mean value of X the statistical average of X
3.1 Expectation E [ X ] = ¯ X = −∞ xf X ( x ) dx : Continuous R.V. E [ X ] = ¯ X = N i =1 x i P ( x i ) : Discrete R.V. N : # of masses (can be ) Ex. 3.1-2 (Exponential distribution) f X ( x ) = λe λx x 0 0 x < 0 E [ X ] = 0 xλe λx dx = 1 λ 0 ze z dz = 1 λ Ex. (Poisson distribution) f X ( x ) = n =0 e λ λ n n ! δ ( x n ) P ( n ) P ( X = n ) = e λ λ n n ! E [ X ] = n =0 ne λ λ n n ! = n =1 e - λ λ n ( n 1)! = e λ k =0 λ k +1 k ! = λ Gaussian, Bernoulli (binomial)? · · ·
3.1 Expected Value of a function of a R.V. Function of a R.V. g : X : function of a R.V. E [ g ( X )] = −∞ g ( x ) f X ( x ) dx : for Continuous R.V. E [ g ( X )] = N i =1 g ( x i ) P ( x i ) : for Discrete R.V. It is known that a particular random voltage can be represented as a Rayleigh random variable V having a density function f X ( x ) = 2 5 xe x 2 / 5 x 0 0 x < 0 The voltage is applied to a device that generates a voltage Y = g ( V ) = V 2 that is equal, numerically, to the power in V (in a 1-ohm resistor) Find the average power in V Power in V = E [ g ( V )] = E [ V 2 ] = 0 2 v 3 5 e v 2 / 5 dv = 5 0 ξe ξ = 5 W
3.1 Expected Value of a function of a R.V. If g ( X ) is a sum of N functions g n ( X ) , n = 1 , 2 , · · · , N , then the expected value of g ( X ) is the sum of the N expected values of the individual functions Conditional Expected Value E [ X | B ] = −∞ xf X ( x | B ) dx E [ g ( X ) | B ] = −∞ g ( x ) f X ( x | B ) dx B : an event. e.g., B = { X b } or B = { X = a } f X ( x | X b ) =